Yes, driving school vehicles do need a Swiss vignette to use Swiss motorways and expressways, just like any other vehicle up to 3.5 tonnes.
Any vehicles and trailers over 3.5 tonnes are subject to the LSVA / PSVA.
Related articles:
What vehicles need a Swiss motorway vignette?
Where can I buy the Swiss motorway vignette?
How much does the Swiss Motorway Vignette cost?